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A video I watched claimed (at 6:50) that while only \$0.25B had been paid for the Louisiana Purchase, the current value was \$1200B (all in 2017 dollars).

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I couldn't find any references, but assuming the latter number is correct, what might it refer to?

The current value of the land?

The estimated value of the land if all structures and roads were removed?

(It seems to me that these numbers would be vastly different, and comparing empty land to one with infrastructure is comparing apples and oranges)

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The video has a transcript with the references. The \$0.25B figure is obtained from here (after adjusting for inflation). Unfortunately, the author does not provide a source for the \$1.2B figure.

However, there are estimates of the value of land elsewhere. For example, here. Their estimates on a map:

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These estimates consider the value of land only, removing the value of structures on that land (which, as you say, makes a big difference; see paper). They use hedonic regression methods to separate land and building values (check from page 6 of document).

I leave to you the task to compute the precise value of the Louisiana Purchase. (you could ask them to send you the data on the map).

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